Electronic states in self-assembled quantum dots with a lens geometry are studied. A conformal analytical map is used to transform the quantum dot boundary into a dot with semi-spherical shape. The Hamiltonian for a carrier confined in the quantum lens is also mapped into an equivalent operator and its eigenvalues and eigenfunctions for the corresponding Dirichlet problem, of confinement by infinite walls, are analyzed. A modified Rayleigh-Schrddinger perturbation theory is developed to obtain analytical expressions for the energy levels and wavefunctions depending on the height b and radius a of the circular cross section of the spherical cap lens. Numerical calculations are shown for typical cases. The effects of decreasing rotational symmetry on the energy states and eigenfunctions of the quantum dot with lens-shape are presented: The degeneracy due to the z-component of the angular momentum m is broken for b ∦ a. Energy states and wavefunctions with m = 0 present the most pronounced influence on the b ∦ a case. Analytical expressions presented here can be used to estimate the sizes of actual self-assembled quantum dots.